Curios by CR in Prime Curios!

Updated on

01 junio, 2018

From

Prime Curios! site http://primes.utm.edu/curios/home.php and my list of curios: http://primes.utm.edu/curios/ByOne.php?submitter=Rivera

No

Number

Details

Key words

1

33 = 32 + 32 + 32

Sum Of Powers

2

3 is the earliest prime p such that p^5 is the sum of 5 consecutive primes: 3^5 = 41 + 43 + 47 + 53 + 59.

Sum Of Csc Primes

3

11

11 is the earliest prime p such that p^3 is the sum of 3 consecutive primes: 11^3 = 439 + 443 + 449.

Sum Of Csc Primes

4

17

17 is the only odd prime p such that p + sod(p) and p - sod(p) are square numbers (52 and 32, respectively).

Sod

5

17

The smallest automorphic prime, i.e., a prime p such that p is the kth prime and p ends in k. [Russo and Rivera]

Automprphic

6

20

The smallest composite number r such that n = 4r + 3 and m = 8r + 7 are primes.

Primes Mode X

7

24

All even numbers up to 24 are expressible as an algebraic sum of the first six odd primes.

Sum Of Csc Primes

8

31

The earliest and the only known case such that the sum of the divisors of two distinct numbers (16 and 25) is the same prime quantity (31), that is to say: 1+2+4+8+16 = 31; 1+5+25 = 31.

Sum Of Divisors

9

39

The least composite odd number that is the sum of the primes between its smallest and largest prime factors (39 = 3 * 13 = 3 + 5 + 7 + 11 + 13).

Sum Of Csc Primes

10

39

Concatenating the 39th row numbers of Pascal's triangle forms a prime.

Pascal Triangle

11

42

The sum of the least consecutive primes ending in "1" (42 = 11 + 31).

Primes Ernding

12

44

The sum of the first emirp pair: 13 + 31.

Emirp

13

46

The least even number 2k such that SOP(2, P(k)) is divisible by 2k, i.e., SOP(2, P(k)) is congruent to 0 mod (2k). Note that SOP(2, P(k)) = the sum of primes from 2 to the prime P(k).

Sum Of Csc Primes

14

48

The smallest even number that can be expressed as a sum of two primes in five different ways (5 + 43, 7 + 41, 11 + 37, 17 + 31, 19 + 29).

Sum In Different Ways

15

49

The smallest composite number whose digits are composite such that the concatenated factors of either digit with the remaining digit(s) is prime.

Concatenate Factors

16

50

Expressible as the sum of primes in two distinct ways such that all primes involved are consecutive (beginning with 2): 50 = 2 + 5 + 7 + 17 + 19 = 3 + 11 + 13 + 23.

Sum In Different Ways

17

61

Prime numbers that end with "61" occur more often than any other 2-digit ending among all primes less than one billion.

Digits Ending

18

67

6^7 is the sum of a twin prime pair.

Twins

19

68

The largest even number that is expressible as the sum of two primes in two different ways (7 + 61, 31 + 37).

Sum In Different Ways

20

76

76 is the only number n that can be partitioned in exactly n ways, using only distinct prime addends.

Partitions

21

107

107 = 2+3*5*7 = 2*3*5+7*11.

Different Artithmetic Expressions

22

233

Describing 233 and repeating the process with each new term produces five more primes, i.e., "one 2, two 3's," generates 1223, etc.

Self Describing

23

241

The smallest prime p such that p^7 can be written as the sum of 7 consecutive primes. Note that 2 + 4 + 1 = 7.

Sum Of Csc Primes

24

313

313, 313 - 2, 313 - 2*3, 313 - 2*3*5 & 313 - 2*3*5*7 are all prime numbers.

Different Artithmetic Expressions

25

386

386 is composite, but its reversal (683) is prime. Note that the sum of 386 with its reversal 683 (1069) is an emirp!

Emirp

26

709

The smallest prime whose cube is the sum of three prime cubes: 709^3 is 193^3 + 461^3 + 631^3.

Cubes

27

807

807 is the smallest proven coefficient k such that a) k*2^n+1 & k*2^n-1 are never twin primes; b) k*2^n-1 & k*2^(n+1)-1 are never Sophie Germain primes (personal communication from Wilfrid Keller, 20 April 2004).

Twins, Sophie Germain

28

859

The largest prime that cannot be the central number in a 3-by-3 prime magic square.

Magic Square

29

929

The smallest palindromic prime whose cube can be expressed as the sum of three odd cubes: 9293 = 693 + 4473 + 8933.

Palprime, Cubes

30

1019

The smallest prime such that its cube (1058089859) is equal to the sum of a prime and its reversal, i.e., 297594067 + 760495792.

Cubes, Reversal

31

1193

The start of the smallest sequence of ten consecutive emirps.

Csc Emirps

32

1619

A prime whose cube can be expressed as the sum of three prime squares in two different ways, i.e., 16193 is 32 + 249672 + 601692 and 32 + 281632 + 587412

Cubes, Squares

33

1683

1683 is the ONLY number N that can be expressed as a sum in exactly N ways using three distinct primes each time (the order of the primes in the sum is irrelevant).

Different Artithmetic Expressions

34

1741

The smallest prime p such that p^9 is equal to the sum of 9 consecutive primes.

Sum Of Csc Primes

35

2003

200*3 + 1, 20*03 + 1, and 2*003 + 1 are three sets of twin prime pairs. Note that 2003 is the largest prime for which this happens.

Different Artithmetic Expressions

36

3313

The smallest prime such that every digit d in it appears exactly d times.

37

4093

The sum of 4093 and the next consecutive prime can be written as a power of 2.

Powers Of 2

38

5246

In which non-trivial calendar year must a man be born, who lives no more than 100 years, in order that both his age and the calendar year are never simultaneously prime numbers? Answer: In the year 5246. This is the earliest year this happens.

Year Of Birth

39

5557

2 + 3 + 5 + 7 + 11 + ... + 3833 = 3847 + 3851 + ... + 5557.

Sum Of Csc Primes

40

6569

The smallest prime p such that p^11 is equal to the sum of 11 consecutive primes.

Sum Of Csc Primes

41

11909

1909 is the smallest prime that divides just one of the 9! zero-less pandigitals, i.e., 142675893. Discovered by Emmanuel Vantieghem.

Pandigitals

42

11927

The concatenation of the first 1429 prime numbers, from 2 to 11927, is a 5719-digit prime. Discovered by Yves Gallot in 1998.

Sum Of Csc Primes

43

13789

The largest prime number with distinct digits in Spanish alphabetical descending order (Uno, Tres, Siete, Ocho, Nueve).

Alphabetical Order

44

25583

Paul Erdos lived from 26/03/1913 to 20/09/1996, or 25 days, 5 months, 83 years

Year Of Birth

45

30103

The sum of 37 consecutive primes (30103 = 683 + 691 + 701 + ... + 937 + 941).

Sum Of Csc Primes

46

34421

The smallest prime expressible in six distinct ways as the sum of consecutive primes.

Sum Of Csc Primes

47

44417

All the primes from 2 to 44417 divides at least one zeroless pandigital. The next prime (44449) is the smallest prime that does not divide at least one zeroless pandigital number.

Sum Of Csc Primes, Pandigitals

48

52583

Paul Erdos lived from 26/03/1913 to 20/09/1996, i.e., 5 months, 25 days and 83 years.

Year Of Birth

49

71317

A palindromic prime that can be expressed as the sum of consecutive primes in three ways.

Sum Of Csc Primes, Palindrome

50

73609

The largest prime with distinct digits such that its square (5418284881) consists of digits not included in the prime.

Squares

51

81619

The largest known prime whose (beastly) square is composed of only two different digits.

Squares

52

94397

The largest known prime (emirp) that produces all distinct primes by deleting any one digit.

Emirps, Delete

53

111109

All the primes from 2 to 111109 divides at least one ten-digit pandigital. This is because the next prime (111119) is the smallest prime that does not divide at least one ten-digit pandigital number.

Sum Of Csc Primes, Pandigitals

54

139967

The lesser prime in the smallest twin prime pair for which the sum is a seventh power: 67. Note that this number ends with 6 and 7. [Rivera and Trotter]

Primes, Powers

55

144319

144319 is the quantity of distinct primes involved in the whole set of zero-less pandigitals. Discovered by Emmanuel Vantieghem.

Pandigitals

56

293339

293339 is the smallest prime that divides just one of the 10! ten digits pandigitals, i.e., 1795234680. Discovered by Emmanuel Vantieghem.

Pandigitals

57

843753

843753 x 222222 ± 1 is a Twin prime pair.

Twins

58

1010203

1010203 = 100^1+100^0+100^1+100^0+100^2+100^0+100^3 is prime.

Sum Of Powers

59

1059002

1059002 "nines" preceeded by one "1" is a near repdigit prime number, found by Serge Batalov in September 2013.

Near Repdigit

60

1102173

1102173 is the quantity of distinct primes involved in the whole set of ten digits pandigitals. Discovered by Emmanuel Vantieghem.

Pandigitals

61

12422153

Replacing each digit d with d copies of the digit d produces another prime throughout three transformations. Confirmed by P. Jobling to be the smallest prime of this type.

Repleacement

62

12815137

If you repleace each digit of an initial prime as 12815137 by its square, you obtain a new prime and up to six primes doing this recursively: 12815137 -> 14641251949 -> 116361614251811681 -> 11369361361164251641136641 -> 11936819361936113616425136161193636161 -> 1181936641819361819361193613616425193613611819369361361

Repleacement, Squares, Recursive

63

20772199

The number 20772199 is the smallest integer with the property that the sum of the prime factors of n and the sum of the prime factors of n+1 are both equal to 666:

Sum Of Prime Factors

64

25794165

The smallest integer that generates 9 prime numbers substracting the nine powers of 2 in order: 2, 4, 8, 16, ... 512. The primes generated are 25794163 25794161 25794157 25794149 25794133 25794101 25794037 25793909 25793653, respectively.

Subtracting Powers

65

27045226

There are 27045226 palindromic primes of length 17. [Eibl , Rivera and Roonguthai]

Palindrome

66

40020041

40020041 squared is equal to 1601603681641681, which contains seven squares (greater than 9) embedded: 16,016,036,81,64,16,81. The smallest such prime with this property.

Squares

67

42666479

Is the prime 42666479 easier to remember than 98765*432-1.

Pandigitals

68

60000607

A prime such that the absolute values of 6 - 0000607, 60 - 000607, 600 - 00607, 6000 - 0607, 60000 - 607, 600006 - 07, and 6000060 - 7 are also primes. This is the largest known example.

Largest Example Known

69

61729859

The smaller in a pair of consecutive primes whose sum produce a zeroless pandigital (61729859 + 61729909 = 123459768).

Twins, Pandigitals

70

61799237

The smaller of the smallest pair of twin primes whose sum produce a zeroless pandigital (61799237 + 61799239 = 123598476).

Twins, Pandigitals

71

68956417

The larger known prime of the form P(m^2 + 1) = n^2 + 1 for m = 2015 and n = 8304. There are two smaller known and no others up to n = 2000000.

Largest Example Known

72

72654130

The prime factorization of 72654130 = 2*5*67*108439. Both expressions use only consecutive distinct digits; 0 to 7 for 72654130 and 0 to 9 for 2*5*67*108439. Note that the last expression is pandigital.

Pandigitals

73

92364991

92364991 to the power 22 is composed of 176 nonzero decimal digits.

Powers

74

109739359

The largest prime factor of any nine-digit pandigital.

Pandigitals

75

191416111

Replace any single digit of 191416111 with a 7 and the result is always prime.

Repleacement

76

197072563

4p2 + 1 is prime for p = 197072563. The new prime will produce yet another prime if placed back into the original formula. This iteration can be repeated for a total of 4 new primes.

Recursive

77

246246017

246246017 is the start of a run of 51 consecutive prime numbers such the last three digits of each is also a prime number. This is the largest run before 2^32.

Consecutive Primes

78

247715873

The smallest semiprime such that its factors taken together (10457*23689) is pandigital.

Semiprimes, Pandigitals

79

253124999

The smaller of the smallest Twin prime pair for which the sum is a 4th power (sum = 1504). [Rivera and Trotter]

Twins, Powers

80

314267743

A first occurrence of eight consecutive primes beginning and ending with three.

Consecutive Primes

81

317130731

The start of the smallest set of five consecutive prime numbers such that each term is the sum of the previous term plus its sum of digits.

Consecutive Primes

82

333667001

The smallest prime that is equal to the sum of the cubes of its third parts: 333^3 + 667^3 + 001^3.

Sum Of Cubes

83

351242153

The five Olympic Rings represent the five major regions of the world: Africa, the Americas, Asia, Europe, and Oceania. 351242153 is a palindromic prime formed from the first five digits. Note that each circular region (http://primes.utm.edu/curios/includes/gifs/351242153.jpg) contains an equal sum of digits.

Palprime, Olympic Circle

84

375656573

The smallest palindromic prime that is also a Sierpinski number

Palprime, Sierpienski

85

493756181

The smaller of the largest pair of twin primes whose sum produce a zeroless pandigital (493756181 + 493756183 = 987512364).

Twins, Pandigitals

86

493826699

Smaller of the largest pair of consecutive primes whose sum produce a zeroless pandigital (493826699 + 493826713 = 987653412).

Csc Primes, Pandigitals

87

511729877

The smaller of the smallest pair of consecutive primes whose sum produce a ten digits pandigital (511729877 + 511729891 = 1023459768).

Csc Primes, Pandigitals

88

511742897

The smaller of the smallest pair of twin primes whose sum produce a ten digits pandigital (511742897 + 511742899 = 1023485796).

Twins, Pandigitals

89

533583823

The smallest number n such that n and 05n + 33n + 58n + 38n + 23n is prime for n = 0 to 10.

Substrings

90

540298673

The largest prime number with distinct digits in Spanish alphabetical order (Cinco, Cuatro, Cero, Dos, Nueve, Ocho, Seis, Siete, Tres).

Alphabetical Order

91

589234176

589234176 (=2^16*3^5*37) has the maximal quantity of prime factors (22) of the zeroless pandigital numbers. No other numbers of this type has 22 prime factors.

Pandigitals

92

608844043

The values of + 608844043, 6 + 08844043, 60 + 8844043, 608 + 844043, 6088 + 44043, 60884 + 4043, 608844 + 043, 6088440 + 43, and 60884404 + 3 are primes. No larger prime with this property is known.

Largest Example Known

93

723121327

Choose any digit d of this number; there is always a matching d spaced d digits apart from it (either to the left or the right), and none of these interceding digits are d. There is no smaller prime for which this is true.

Matching

94

725638914

The smallest zeroless pandigital with 8 distinct prime factors (2*3^2*7*11*13*17*23*103). There are 3 other zero-less pandigital with this property (789256314, 856197342, and 961327458). No zeroless pandigital has more than 8 distinct prime factors.

Pandigitals

95

933739397

Any substring from the right of this prime number is prime.

Substrings

96

1097393447

The largest prime factor of any pandigital number: 9876541023 = 9*1097393447.

Pandigitals

97

1113443017

The first in a sequence of two dozen consecutive primes of the form 4n + 1.

Csc Primes 4N+1

98

1304119699

Replace any single digit of 1304119699 with a 7 and the result is always prime.

Repleacement

99

1364103977

The smallest prime such that (1364103977 mod p) is prime for the first 17 primes p after 2 (3, 5, 7, ... , 61).

Integers Mod P

100

1654792830

1654792830 is the smallest pandigital with 26 (maximal) distinct primes embedded, reading in both directions.

Pandigitals

101

1966640443

The only known prime whose index (it is the 96664044th prime) is a substring (between its first and last digits). Note that the beast number is lurking inside.

Prime Index

102

2050918644

2050918644 is the first member of the earliest 6-tuple of consecutive Smith (composite) numbers, and the largest known k-tuple as of 19 December 2003.

Largest Example Known

103

2113733797

2113733797 is the smallest integer having thirty embedded (distinct) primes.

Substrings

104

2148736590

2148736590 is the smallest ten digits pandigital with 9 distinct prime factors (2*3^2*5*7*11*13*17*23*61).

Pandigitals

105

2462372461

Starting at 2462372461, there are fifty-seven consecutive primes not ending in the digit "3". This is the largest run less than 2^32.

Csc Primes

106

3215031751

The strong k-pseudoprime test for k = 2, 3, 5, 7, correctly identifies all primes below 2.5 x 10^10 with only one exception, i.e., 3215031751 = 151*751*28351.

Exception, Strongseudoprimes

107

3781378039

3781378039 is the largest prime number p such that p - q# are primes for all 2 < q< p, that is to say, all these are primes: 3781378039, 3781378039 - 2, 3781378039 - 2*3, 3781378039 - 2*3*5, ..., 3781378039 - 2*3*5*7*11*13*17*19*23.

Primorials

108

4076863487

The first of the smallest twin prime pair that adds to a fifth power (96^5).

Twins, Powers

109

4263918750

The largest of only two pandigital numbers (the other is 3912657480, found by Federico di Francesco) having, for any pandigital, the largest quantity (24) of embedded divisors. In this case, the divisors for 4263918750 are: 1, 2, 3, 5, 6, 7, 9, 42, 26, 63, 39, 91, 18, 75, 50, 875, 750, 1875, 8750, 91875, 18750, 918750, 426391875, and 4263918750.

Pandigitals

110

4938256511

The smaller of the largest pair of twin primes whose sum produce a ten digits pandigital (4938256511 + 4938256513 = 9876513024).

Twins, Pandigitals

111

4938271579

The smaller of the largest pair of consecutive primes whose sum produce a ten digits pandigital (4938271579 + 4938271631 = 9876543210).

Twins, Pandigitals

112

6398410752

The pandigital number containing the largest quantity of prime divisors (6398410752=2^21*3^3*113).

Pandigitals

113

6650672641

The largest prime such that its k left-most digits are divided by the prime p(k), for k=1 to 9. E.g., 2 divides 6, 3 divides 66, 5 divides 665, etc.

Leftmost Digits

114

6829547103

6829547103 & 8563421790 are the only two pandigital numbers that have 20 primes embedded. No other pandigital has more than 20.

Primes Embedded, Pandigitals

115

8970134652

The largest pandigital whose reversal and itself are respectively sandwiched between twin primes.

Pandigitals, Reversal, Twins

116

11172427111

The smallest palindromic prime such that the square of the sum of its digits equals the product of its digits. [Rivera and De Geest]

Paprimes, Sod & Pod

117

12456783197

The smallest prime that contains all the prime numbers from 2 to 97 as subsequences.

Subsequences

118

26404836711

26404836711 produces exactly 93 primes in a Collatz trajectory. Note that this may not be the smallest case that produces 93 primes. See Puzzle 634.

Collatz Trajectory

119

61605403707

61605403707 is the only integer composed of 11 digits that produces a prime number by inserting orderly prime numbers starting with the prime 2 (except the prime 3), one at a time, in every position. Discovered by Jan van Delden (May 2018): 61605403707 -> {261605403707, 651605403707, 617605403707, 6161105403707, 6160135403707, 6160517403707, 6160541903707, 6160540233707, 6160540329707, 6160540373107, 6160540370377, 6160540370741}

Insertion

120

351725765537

The concatenation of the five known Fermat primes 3, 5, 17, 257, and 65537. Concatenate them backwards, 655372571753, and it remains prime.

Fermat Primes, Concatenation

121

411379717319

The smallest-largest prime such that every two consecutive digits produce a distinct prime number: 41, 11, 13, 37, 79, 97, 71, 17, 73, 31 and 19, a total of eleven primes. No more than eleven distinct primes are possible this way.

Pair Of Consecutive Digits

122

1835211125381

The smallest palindromic prime that is also a Riesel number.

Palprime, Riesel Number

123

18285670562881

The only known emirp to be formed by concatenating a row of Pascal's triangle.

Emirp, Pascal Triangle, Concatenation

124

28862180229031

1+2^34*5*6*7*8-90 is a prime number.

Pandigital Expresions

125

28862180229121

1+(2^34)*5*6*7*8 is a prime number.

Pandigital Expresions

126

55555544444441

55555544444441 is the smallest prime that produces a chain of 5 primes applying the following rule to each prime of the chain: Sum every digit powered to itself. E.g., 55555544444441 -> 20543 -> 3413 -> 311 -> 29. A chain of 6 provable primes will not be established for a long, long time.

Largest Example Known

127

578415690713087

The smaller of the smallest Twin prime pair for which the sum is a 6th power (sum = 3246). [Rivera and Trotter]

Twins, Powers

128

1990474529917009

The smallest prime formed from the reversal of a composite Mersenne number (2^53-1).

Reversal, Mersenne

129

1023976197718878397

1023976197718878397 = 7*7*7*7*61*89*971*983*82301 is believed to be (as of October 2013) the smallest known nine-prime-factors composite number, containing the same digits as the set of its prime factors. Reported by J. K. Andersen.

Sod

130

1111111111111111111

The smallest prime Kaprekar number. E.g., 1111111111111111111^2 = 1234567901234567900987654320987654321 and 123456790123456790 + 0987654320987654321 = 1111111111111111111.

Kaprekar Prime Number

131

1111918171614121013

The largest self-inventoried prime, i.e., it says itself ("eleven 1s, one 9," etc.).

Self-Inventoried

132

71322723161814151019

The largest self-descriptive prime without repetition.

Self-Descriptive

133

73099303486215558911

The smaller of the smallest Twin prime pair for which the sum is a 9th power, i.e., 1749. [Rivera and Trotter]

Twins, Powers

134

12360...09013

1236 0600901222 5672009013, A prime number composed of exactly two dozen digits where the first k digits (from the left) are divided by k, for k=1 to 23.

Leftmost Digits

135

36085...36719

The average of 3608528850368400786036719 and the next prime (3608528850368400786036731) is 3608528850368400786036725, which is the largest integer such that its k leftmost digits are divided by k, for k=1 to 25. [Rivera]

Leftmost Digits

136

53583...71840

 53583 5925499096 6640871840 is The smallest Fibonacci number divisible by the first five primes.

Fibonacci

137

14097...72287

140975 6730907423 9886172287 is The smaller of the smallest Twin prime pair for which the sum is a 8th power, i.e., 1518^8.

Twins, Powers

138

48256...13571

482564152712479922509389813571 is a prime p such that p^2 contains the next prime after p, 482564152712479922509389813577; p^2=23286816148311364375316645(482564152712479922509389813577)2041. Discovered by Giovanni Resta in May 2018.

Primes Embedded, Squares

139

99999...23343

9999 9999988888 8887777777 6666665555 5444223343 IS The largest prime p such that every decimal digit d appears exactly d times.

Pandigitals

140

31579...37111 (396 digits)

31579...37111
2^2703 is an 814-digit integer. If we drop all of the even digits, a 396-digit prime (31579...37111) remains.

Dropping Digits

141

10000...18919 (1000 digits)

10000...18919
The smallest titanic prime p and Polignac number, i.e., not of the form p = q + 2^x, where q is prime.

Titanic, Polignac

142

13138...81469 (1000 digits)

13138...81469
A titanic prime number (2^3319 + 4073181) that remains prime after each even digit is switch to the next odd digit, i.e., 0->1, 2->3, etc.

Titanic, Converting Digits.

143

19999...37601 (1001 digits)

1 9999...37601
The first known titanic example of an oblong number that is the sum of 2 successive primes.

Titanic, Oblong, Sum Of Csc Primes